\ast$$\end{document}b^* for regression dilution using SIMEX (in data) or Hedges–Olkin adjustment (in simulations), obtaining the corrected slope estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b$$\end{document}b^.For each SNP, the bias-corrected estimate of its effect on prognosis is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \beta _{GY} = \hat \beta _{GY}^\prime - \hat b\hat \beta _{GX}$$\end{document}β^GY=β^GY′-b^β^GX with standard error \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt {\sigma _{GY}^2 + \hat b^2\sigma _{GX}^2 + \hat \beta _{GX}^2var\left( {\hat b} \right) + \sigma _{GX}^2var\left( {\hat b} \right)}$$\end{document}σGY2+b^2σGX2+β^GX2varb^+σGX2varb^.Hypothesis tests and P-values for each SNP may be computed by referring the ratio of the adjusted estimate and its standard error to the standard normal distribution.
